Problem: Simplify and expand the following expression: $ \dfrac{-10}{4y - 8}+\dfrac{4y - 7}{y - 4} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4y - 8)(y - 4)$ Multiply the first term by $\dfrac{y - 4}{y - 4}$ $ \begin{align*} \dfrac{-10}{4y - 8} \times \dfrac{y - 4}{y - 4} & = \dfrac{(-10)(y - 4)}{(4y - 8)(y - 4)} \\ & = \dfrac{-10y + 40}{(4y - 8)(y - 4)}\end{align*} $ Multiply the second term by $\dfrac{4y - 8}{4y - 8}$ $ \begin{align*} \dfrac{4y - 7}{y - 4} \times \dfrac{4y - 8}{4y - 8} & = \dfrac{(4y - 7)(4y - 8)}{(y - 4)(4y - 8)} \\ & = \dfrac{16y^2 - 60y + 56}{(y - 4)(4y - 8)}\end{align*} $ Now we have: $ = \dfrac{-10y + 40}{(4y - 8)(y - 4)} + \dfrac{16y^2 - 60y + 56}{(y - 4)(4y - 8)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{-10y + 40 + 16y^2 - 60y + 56}{(4y - 8)(y - 4)} $ $ = \dfrac{-70y + 96 + 16y^2}{(4y - 8)(y - 4)}$ Expand the denominator: $ = \dfrac{-70y + 96 + 16y^2}{4y^2 - 24y + 32}$ Simplify: $ = \dfrac{-35y + 48 + 8y^2}{2y^2 - 12y + 16}$